Stability, corners, and other 2-dimensional shapes

Abstract: We introduce a relaxation of stability, called robust stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a pseudofinite group. We show that robust stability satisfies a stationarity principle for measure independent elements. We apply this principle to deduce the existence of squares in dense robustly stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products of finite groups. In the final section, we establish the existence of $3\times 2$-grids (and thus of $L$-shapes) in dense robustly stable $2$-dimensional subsets of finite abelian groups of odd order.